The tangent space is the intrinsic linearization of a manifold at one point. It can be defined through curves, derivations, or coordinates; the equivalence of these descriptions is what makes geometric and analytic calculations agree.
Derivations and coordinate vectors
Definition. A tangent vector at \(p\in M\) is a derivation \(v:C^\infty_p(M)\to\mathbb R\): it is linear and satisfies \[ v(fg)=f(p)v(g)+g(p)v(f). \] The vector space of derivations is \(T_pM\).
Theorem. In a chart \((x^1,\ldots,x^n)\), every \(v\in T_pM\) has a unique representation \[ v=\sum_{i=1}^n v(x^i)\left.\frac{\partial}{\partial x^i}\right|_p. \] Consequently \(\dim T_pM=n\).
Proof. For a smooth \(f\), Hadamard's lemma in coordinates gives \[ f(x)-f(p)=\sum_i(x^i-x^i(p))g_i(x),\qquad g_i(p)=\partial_if(p). \] Apply the derivation \(v\). Constants vanish and the Leibniz rule gives \(v(f)=\sum_i v(x^i)\partial_if(p)\), proving existence and uniqueness.
If \(\gamma:(-\varepsilon,\varepsilon)\to M\) satisfies \(\gamma(0)=p\), then \[ \dot\gamma(0)f=\left.\frac{\mathrm{d}}{\mathrm{d}t} f(\gamma(t))\right|_{t=0} \] is a derivation. In coordinates, every tangent vector is represented by the curve \(t\mapsto\varphi^{-1}(\varphi(p)+tv)\).
The tangent bundle
Definition. The tangent bundle \[ TM=\coprod_{p\in M}T_pM \] is a \(2n\)-manifold with induced coordinates \((x^1,\ldots,x^n,v^1,\ldots,v^n)\). The projection \(\pi:TM\to M\) sends \(v\in T_pM\) to \(p\).
Under a coordinate change \(y=y(x)\), fiber coordinates transform by \[ w^a=\sum_i\frac{\partial y^a}{\partial x^i}v^i. \] This linear transformation law is the prototype for vector bundles.
Differentials and the chain rule
Definition. For a smooth map \(F:M\to N\), its differential at \(p\) is \[ \mathrm{d}F_p:T_pM\to T_{F(p)}N,\qquad (\mathrm{d}F_pv)(h)=v(h\circ F). \]
Proposition. If \(F:M\to N\) and \(G:N\to P\) are smooth, then \[ \mathrm{d}(G\circ F)_p=\mathrm{d}G_{F(p)}\circ\mathrm{d}F_p. \] In coordinates, \(\mathrm{d}F_p\) is represented by the Jacobian matrix.
Proof. For \(h\in C^\infty(P)\), \[ \bigl(\mathrm{d}(G\circ F)_pv\bigr)h =v(h\circ G\circ F) =\bigl(\mathrm{d}G_{F(p)}(\mathrm{d}F_pv)\bigr)h. \]
Cotangent vectors
The dual space \(T_p^*M\) has basis \((\mathrm{d}x^1,\ldots,\mathrm{d}x^n)\), where \(\mathrm{d}x^i(\partial/\partial x^j)=\delta^i_j\). For \(f\in C^\infty(M)\), \[ \mathrm{d}f_p=\sum_i\frac{\partial f}{\partial x^i}(p)\,\mathrm{d}x^i. \] The differential is naturally a covector; a gradient vector appears only after a metric identifies \(T_pM\) with \(T_p^*M\).